Place warm water in large bowl. Sprinkle yeast and 1/4 teaspoon sugar over and stir to blend. Let stand until yeast dissolves and mixture is slightly foamy, about 8 minutes.
Whisk milk, 1 egg, salt and remaining 1/3 cup sugar into yeast mixture. Add melted butter and whisk until smooth. Add 1 cup flour and mix until smooth. Combine 4 cups flour and 1 cup chilled butter in processor. Using on/off turns, process until mixture resembles coarse meal. Add to yeast mixture and stir until dry ingredients are moistened. Knead in bowl until smooth dough forms, adding more flour if dough is sticky, about 5 minutes.
Cover bowl with plastic wrap. Refrigerate dough overnight. (Can be prepared 2 days ahead. Keep refrigerated.).
Butter twenty-four 1/3-cup nonstick muffin cups. Turn dough out onto floured surface; knead briefly until smooth and elastic, about 3 minutes. Divide dough into 4 equal portions. Place 1 dough portion on work surface; cover and chill remaining dough. Roll out 1 dough portion on floured surface to 13x11-inch rectangle (about 1/8 inch thick). Cut rectangle lengthwise into 6 strips, each scant 2 inches wide. Stack strips atop one another, forming 6 layers and pressing slightly to adhere. Cut strips crosswise into 6 equal stacks, each about 2 inches long. Place 1 dough stack, 1 cut side down, into each muffin cup (dough will fan out slightly and fill muffin cups as dough rises). Repeat with remaining chilled dough pieces.
Cover rolls with kitchen towel. Let rise in warm draft-free area until rolls are puffed and doubled in volume, about 1 hour and 15 minutes.
Position 1 rack in center and 1 rack in top third of oven; preheat to 350°F Brush rolls gently with egg glaze. Sprinkle with poppy seeds. Bake until rolls are golden brown, switching top and bottom baking sheets halfway through baking, about 25 minutes. Transfer pans to rack and cool rolls 5 minutes. Remove rolls from pans and cool on racks. Serve warm or at room temperature. (Can be prepared 1 week ahead. Wrap in aluminum foil and freeze. If desired, rewarm thawed wrapped rolls in 350°F oven about 10 minutes.).